power_tail_integral

std.elementwise.power_tail_integral · Level L1

The improper integral ∫_lower^∞ scale/x^p dx for lower > 0, defined as the limit of the integrals up to b as b → ∞: it converges exactly when p > 1, to scale·lower^(1 − p)/(p − 1) (0 is returned when it diverges). Integrals up to finitely many cut-offs, however stable they look, do not show convergence.

∫_ℓ^∞ s·x^(−p) dx = s·ℓ^(1−p)/(p − 1) for p > 1

Signature

power_tail_integral(scale: f64[], p: f64[], lower: f64[]) → f64[], f64[]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Agrees with the reference the closed form when p > 1 to 80 digits (100-digit arithmetic), on all 40 test cases.
All 80 float64 results inside the running error bound; the closest uses 11% of it.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them inside the running error bound at u = 2⁻²⁴; the closest uses 12% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 89%
bit-equal to the NumPy formula in float64: 85%
largest error, in units in the last place: 7.71

Identity

Calls

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Called by

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sha256:ab78cde52ac121783cff875567d39dbaa2b90cc1cdcdcadbd39b2ea785d6aa51

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.elementwise.power_tail_integral · Ω:power_tail_integral
Pins: sha256:625ccc02c92b2c2ce3df8b5a240696d20bdfa5db6e41d8cdbdeb5051a3f59a79this graph alone
Evidence: sha256:3c15193b6332319e845c9a16cad51a7bf2ae116226dc5fb6fc268c1ce3bf229bthe hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.